According to Sir IsaacNewton, the
resultant force vector on an object of mass M at a given momentis M times the acceleration.In Chapter 4, we explored this law with
many examples in which the acceleration and force were constant. In this
experiment you will test the theory for the special case of uniform
circular motion. As discussed inChapters5 and 8,the acceleration , force and velocity
magnitudes are constant ; only the direction of theaccelerationandvelocity changes, that’s it.

The force causing this special motion is called the
centripetal force , or “center seeking” force. It points to the circle’scenterand hasconstant magnitude F =
mv2/r, where v is the speed ( i.e. velocity magnitude) and r is
theradius. It is not a new force,
itis only a namewe assign to any force that causes
uniformcircularmotion, whether it be the gravitational and
electricalforceunderlyingthe Earth’s orbitaround the
Sunandthe electron’s motion around the
nucleus,respectively.In the example of a rocktwirled in a vertical circle, the
centripetalforce at any point is a
combination of gravitational and tension forces.See chapter5.

To test the theory, you will measure the centripetal force
magnitude F by two methods , static and dynamic:

(A)A certain force F acting on the mass M is measured
under static conditions by a weight mg.We could call this the static value of F.

(B)When the mass under goes uniform circularmotion this force should be equal to the
centripetal force, and thus equal to Mv2/r.Thus we are testing whether the static and dynamic values are equal :

mg=Mv2/r.(1)

The experiment can easily be illustrated with a free body
diagram provided by the instructor-- See figure 1, page 3,of the experimental set up described here:

Let’s examine thedynamic measurement
(B). As the bob of mass M moves in a horizontal circle of radius r, the
spring stretches and the vertically hanging string sweeps out a cylindrical
surface whose sides are parallel to the central axis of rotation.The spring force magnitude Fs ,
exerted on the bob when the spring stretches,is automatically included in the both
dynamic (B) and static measurements (A) of F.
In the dynamic case, thetotal
centripetal force has magnitude F = Fs , orFs=Mv2/r.In this derivation we assume that the
spring connecting the bob to the central rodis horizontal--as in the static case (A) .

(A) STATIC MEASUREMENT. Make
sure the axle-to-bob spring and the bob-to-pulley string areprecisely horizontal by measuring the
height above the table at both ends, and then raising and lowering the pulley
until the heights are equal.You can
make the platform parallel with the table by adjusting the clamps and knobs.

(Q-1) Make the static measurement of F taking 5 readings.(See the Picket Fence Free Fall lab handout
as a reference. ) Here is why we make multiplereadings. It’strue that one way to state the precision of
our readingsis to assume anuncertaintyΔm inst= 1 g for each
“simple” measurement;weassumethe “instrument” uncertaintyis
determined only by the measurement technique. This means for m, written on grams (g),the least significant digit is in the
one’splace with a 1 guncertainty.Thisuncertainty results from just measuring the mass needed to bring the
bob pointer in alignment with the platform pointer.

But the measurementmay be
“non-simple”---it may have a larger uncertaintythan the technique uncertainty. There may
be other randomerrors besides those
from just reading the mass after aligning the pointer. To cover this
possibility,we thus make the static
measurement of F taking 5 readings. In this case, westate the precision by takingthe difference between the minimum and
maximum valuesand dividing by N
(where N = 5).We call this quantity
R/N.Compute R/N .Compare this approach with the Picket Fence
Free Fall lab.Compute the average (mBEST)
of the 5 measured masses m.The
uncertaintyyou choose will either
beΔm inst or R/N ,
which ever is larger.In either case,
the uncertainty is rounded to just one digit and we round the average
(“best”) value to the same decimal place. As in the picket fence lab, the
result is reported as
mBEST ± uncertainty .

Also measure r, the distance
between the axis of rotation to the center of the bob.Take 5 readings.Again, we can state the precision of our
readings by assumingonly an
“instrument” uncertaintydue torandom errors from just reading the
instrument.We will assume that the
accuracy of any single reading with a meter stick is plus or minus ¼ of the
smallest scale division of0.1 cm. See
page 36 here: http://dept.physics.upenn.edu/~uglabs/lab_manual/Lab_Manual.pdf

Thus , Δrinst =0.1/4 = 0. 025 = 0.03 for a single reading
witha meter stick . But the length r
is measured from a difference of labeled marks on the stick.It can be shown that a more accurate “instrument”
uncertainty is times the
uncertainty for a single reading. See pages 49-50 here: http://dept.physics.upenn.edu/~uglabs/lab_manual/Lab_Manual.pdf

Δrinst =·(0.1/4) = 0. 03525 = 0.04.(2)

But justin case themeasurements are “non-simple”-- make the r measurementtaking 5 readings. Takethe difference between the minimum and
maximum valuesand divide by N (where
N = 5) . This result is calledR/N.Compute the average (rBEST)
of the 5 measured radii.The
uncertaintyyou choose will either
beΔr inst or R/N ,
which ever is larger.In either case,
the uncertainty is rounded to just one digit and we round the average
(“best”) value to the same decimal place. As in the picket fence lab, the
result is reported as rBEST ± uncertainty .

(B)DYNAMIC MEASUREMENT . To measure the dynamic value of
the centripetal force, you need to measureMv2/r.This is done by measuring r (see Q-1)and the period T.
(Q-2) Derive the following expression for the dynamic value of the
centripetal force in terms of the measured quantities M, r and T:

F = 4π2Mr/T2. (3)

(Q- 3) Measure M:Again, we can state the precision of our
readings by assumingonly an
“instrument” uncertaintydue torandom errors from just reading the
instrument.We will assume that the
accuracy of any single reading with a mass scaleis plus or minus ¼ of the smallest scale
division of0.1 g. See page 36 here: http://dept.physics.upenn.edu/~uglabs/lab_manual/Lab_Manual.pdf

Thus , ΔMinst =0.1/4 = 0. 025 = 0.03 for a single reading
witha mass scale.But the measurementmay be “non-simple . We thus make the M
measurementtaking 5 readings. In this
case, westate the precision by takingthe difference between the minimum and
maximum valuesand dividing by N
(where N = 5.)We call this quantity
R/N.Compute R/N .Compute the average (MBEST) of
the 5 measured masses. The uncertaintyyou choose will beeitherΔM inst or R/N , which ever
is larger.In either case, the
uncertainty is rounded to just one digit and we round the average (“best”)
value to the same decimal place. The result is reported as MBEST ±
uncertainty .

(Q-4)MeasureT : Again, we can state
the precision of our readings by assumingonly an “instrument” uncertaintydue torandom errors from just
reading the instrument.We will assume
that the accuracy of any single reading with a digital timerscaleis plus or minus 0.001 s or 0.0001 s, depending on its setting. Because
you are defining T from trials involving 25 rotations, then the
"instrument" uncertainty is reduced by a factor of 1/n = 1/25, where
n = 25 is the number of trials

ΔTinst =0.001 s/
n or
0.0001 s / n (4),

depending on its setting.

In this experimental , the timer setting was for 0.0001 s. But the measurementmay be “non-simple.” We thus make the T
measurementin10 readings. In this case, westate the precision by takingthe difference between the minimum and
maximum valuesand dividing by N
(where N =10.)We call this quantity
R/N.Compute R/N .Compute the average (TBEST) of
the 10 measured times.Each
trial is based on rotating the system 25 times. The uncertaintyyou choose will be eitherΔT inst or R/N , which
ever is larger.In either case, the
uncertainty is rounded to just one digit and we round the average (“best”)
value to the same decimal place. The result is reported as TBEST ±
uncertainty .

(Q-5) Using (Q–1), (Q-3 ) and
(Q-4), compute the dynamic value ofFBEST
using equation(3). For the value of r
, M and T use the “best” value computed in(Q–1), (Q-3 ) and (Q-4)

(Q-6) .Beginningwithequations (3), it is easy
to derivewithMath 1 and 3 calculus
the expression for the uncertainty in the dynamic value of the centripetal
force you computed in the previous part:

The expression contains the “best “ values from (Q–1), (Q-3 ), (Q-4) and
(Q-5).Δr,ΔM and ΔT are the
uncertainties in (Q–1), (Q-3 ) and (Q-4).

(Q-7)Apply the following test to determine
whether the data from this experiment supports the
the hypothesis idea that thestatic
and dynamicmeasurement yield
identical results:
|FBEST – mBEST··g|< 2(ΔF + Δmg), where Δm is the uncertainty in the
hanging mass (Q-1) bringing the
pointer into alignment withbob. This
test will be discussed in further detail.